Motivated by the goal of understanding the evolution of populations undergoing selection, we consider branching Brownian motion in which particles independently move according to one-dimensional Brownian motion with drift, each particle may either split into two or die, and the difference between the birth and death rates is a linear function of the position of the particle. We show that, under certain assumptions, after a sufficiently long time, the empirical distribution of the positions of the particles is approximately Gaussian. This provides mathematically rigorous justification for results in the biology literature indicating that the distribution of the fitness levels of individuals in a population over time evolves like a Gaussian traveling wave.
MR was supported by a Royal Society University Research Fellowship, and JS was supported in part by NSF Grant DMS-1707953.
Both authors thank Jiaqi Liu for spotting a typo in an earlier version of the manuscript, and Daniel Fisher for bringing to their attention the references [16, 19, 20]. They also thank two referees for helpful comments that improved the exposition of the paper.
Matthew I. Roberts. Jason Schweinsberg. "A Gaussian particle distribution for branching Brownian motion with an inhomogeneous branching rate." Electron. J. Probab. 26 1 - 76, 2021. https://doi.org/10.1214/21-EJP673